Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

# Exponentials and Logarithms Properties in an Extended Complex Number Field

Version 1 : Received: 6 April 2021 / Approved: 7 April 2021 / Online: 7 April 2021 (15:20:39 CEST)
Version 2 : Received: 11 April 2021 / Approved: 12 April 2021 / Online: 12 April 2021 (14:31:00 CEST)
Version 3 : Received: 19 September 2021 / Approved: 20 September 2021 / Online: 20 September 2021 (12:09:28 CEST)
Version 4 : Received: 28 November 2021 / Approved: 29 November 2021 / Online: 29 November 2021 (11:11:55 CET)

How to cite: Tischhauser, D. Exponentials and Logarithms Properties in an Extended Complex Number Field. Preprints 2021, 2021040207 (doi: 10.20944/preprints202104.0207.v2). Tischhauser, D. Exponentials and Logarithms Properties in an Extended Complex Number Field. Preprints 2021, 2021040207 (doi: 10.20944/preprints202104.0207.v2).

## Abstract

In this study we demonstrate the complex logarithm and exponential multivalued results and identity failures are not induced by the exponentiation and logarithm operations, but are solely induced by the definition of complex numbers and exponentiation as in C. We propose a new definition of the complex number set, in which the issues related to the identity failures and the multivalued results resolve. Furthermore the exponentiation is no longer defined by the logarithm, instead the complex logarithm formula can be deduced from the exponentiation. There is a cost as some algebraic properties of the addition and substraction will be diminished, though remaining valid to a certain extent. Finally we attempt a geometric and algebraic formalization of the new complex numbers set. It will appear clearly the new complex numbers system is a natural and harmonious complement to the C field.

## Keywords

Complex number field; Complex exponentiation; Complex logarithm; Exponential and logarithm identities

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

Comment 1
Commenter: Daniel Tischhauser
Commenter's Conflict of Interests: Author
Comment: We fixed writing errors in the result sections and, for clarity purpose, added or rephrased some comments
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